Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498821 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,13)(9,24)(10,26)(11,14)(15,22)(16,23)(17,29)(18,36)(19,34)(20,28)(21,30)(25,32)(27,31)(33,35);(1,2)(3,4)(5,7)(6,12)(8,21)(9,17)(10,27)(11,20)(13,30)(14,28)(15,33)(16,34)(18,25)(19,23)(22,35)(24,29)(26,31)(32,36);(1,3,4,2)(6,13)(7,17)(8,22)(9,25)(10,23)(14,24)(15,16)(18,19)(20,21)(26,32)(27,35)(29,36)(30,33)(31,34);(6,14)(12,28)(13,24)(15,32)(16,26)(29,30)(31,34)(33,36)Polycyclic presentation:
2^9|c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322357, #72322361, #72322369, … |
| maximal normal subgroup of | #72322357, #72322361, #72322369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406269);